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Thursday, 18 February 2010

Maths is so arbitrary

I like some maths, but I've always found the basic entity of mathematics a trifle scary. Not because it's difficult, but because it's arbitrary. When you do science, what you say has to be based on the world around us. You can always check things against reality. However, mathematics is an isolated world. You can make a totally arbitrary set of statements, and as long as they are self-consistent, then they're okay. Human beings make up the rules in maths, and that's the scary bit.

Take one simple example - prime numbers. Until the 1800s, the number 1 was mostly considered to be a prime number. After all it has no divisors other than itself and 1 - so it should be. But then it was decided that one was too unique (!) - its strange properties like 1x1=1 made it somehow not a real prime number. So now 1 is excluded from the list and prime numbers start at 2. That's fine, that's the rule - but it is painfully arbitrary. If they wanted to, someone could decide that 2 wasn't a prime number either, as every other prime number is odd. They haven't yet, but it could happen.

Science may be weird and wonderful. And it mostly depends on maths - so some maths I'm very happy with because it is tied into reality. But the work of some pure mathematicians gives me the willies.

9 comments:

It wasn't an arbitrary decision to exclude one from the list of prime numbers. Having it there was causing problems with various bits of number theory about prime numbers, and tests for prime numbers, because of all its arbitrary properties. To get round it, mathematicians chose to use a very precise definition for a prime number and make it a number that has exactly two factors. 1 only has one factor - 1 - and therefore isn't included.

I think part of the beauty of maths is that it can drill down deep into reality and reveal fundamental truths and connections. I embrace the arbitrary, I guess.

The Fundamental Theorem of Arithmetic wants to say that "any integer greater than 1 can be written as a unique product (up to ordering of the factors) of prime numbers". 1 has to be excluded from the primes for this to hold. With 1 out of the way, the FTA generalizes over more-abstract algebraic objects and allows us to talk about when we can expect unique factorization to occur there: for example all complex numbers of the form a+ib*sqrt(5) do not have unique factorization, because 6 = 2*3 = (1+sqrt(5)i)*(1-sqrt(5)i). And then we go on to prove Fermat's Last Theorem...

What the others said. In an integral domain, prime elements cannot be units. Mathematicians got it wrong, first time round. But it is a very minor issue. Science is much more arbitrary. Why is the charge on an electron negative? And do not pull that old one about maths being abstract - every branch of it is inspired by observation of the natural world.

Sorry, Philip, you can't get away with that! Science isn't arbitrary, 'negative' is just a label - but it's applied to a real phenomenon.

What those who are trying to justify the omission of 1 from the primes are doing is explaining why mathematicians chose to be arbitrary, but not showing that this isn't the case.

Here's another example - knot theory. It says, roughly, that a knot is a one-dimensional line, wrapping it around itself in some way, and then having its two free ends fused together to form a closed loop. No that's not a knot. That's a mathematician's arbitrary idea to make the maths work. A knot often hasn't got its ends fused. Try tying your shoe laces with the ends fused.

Both maths and science are fundamentally unnatural, and present concepts that are odds with what we would like to think of as 'common-sense' (see Louis Wolpert).

But the abstractions in maths are harder to reconcile with experience because we have fewer comprehensible metaphors.

Steven Strogatz's lovely comment piece in the NYTimes (http://opinionator.blogs.nytimes.com/2010/02/07/rock-groups/) showed how we can teach basic mathematical metaphors with rocks. But maths is still taught in such a way that we must take equations and their results on trust, so we grow up with a less rich metaphorical-mathematical world.

The metaphors of science are well established, and whilst they describe and predict real and verifiable phenomena, they are still metaphors with an often oblique relation to the physical world.

How can they be otherwise when we know that an electron in a cloud around a nucleus is simultaneously nowhere and everywhere in that cloud, or that the bright colours of a Hubble image bear no relation to what we see with our bare eyes. And yet both metaphorical descriptions help us get a handle on the scientific world.

I'd like to think that if that internal world of mathematical metaphor was cultivated we'd better understand pure maths in real terms. I confess to not yet being there myself...

I would say, though there's a difference between science being counter-common sense and it being unnatural. It is inherently natural, as it describes/explains nature. It's just that common sense isn't very good.

We often use metaphors to explain scientific concepts, though scientists tend to work with models, which are more quantifiable than metaphors.

You are confusing the label "prime" understood through having no divisors and the notion of prime numbers as building blocks of integers (through the fundamental theorem of arithmetic). There are no arbitrary rules - we study the the latter notion because there is more to say about it.

Well, actually there is anonymous - if you'd read the post, it's not me that has changed the definition. And also if prime has two different meanings, it seems to me that it's mathematicians who need to get their act together, not me...

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Author of Science for Life,The Quantum Age, Final Frontier, Dice World, Gravity, The Universe Inside You, Build Your Own Time Machine, Inflight Science, A Brief History of Infinity, The God Effect and more, Brian spends most of his time these days writing popular science books and giving talks.